On Aα-spectrum of a unicyclic graph.

Let G be a graph of order n with adjacency matrix A(G) and diagonal matrix D(G). For any real α ∈ [ 0 , 1 ] , denote A α (G) : = α D (G) + (1 - α) A (G) be A α -matrix of graph G. The eigenvalues of A α (G) are λ 1 (A α (G)) ≥ λ 2 (A α (G)) ≥ ⋯ ≥ λ n (A α (G)) , the largest eigenvalue λ 1 (A α (G)) is called the A α -spectral radius of G. The A α -separator S A α (G) of graph G is defined as S A α (G) = λ 1 (A α (G)) - λ 2 (A α (G)) . For two disjoint graphs G 1 and G 2 (where V (G 1) and V (G 2) are disjoint with v 1 ∈ V (G 1) , v 2 ∈ V (G 2) ); the coalescence of G 1 and G 2 with respect to v 1 and v 2 is formed by identifying v 1 and v 2 and is denoted by G 1 · G 2 . The A α -characteristic polynomial of G is defined to be Φ (A α ; x) = d e t (x I n - A α (G)) , where I n is the identity matrix of size n. A unicyclic graph is a simple connected graph in which the number of edges is equal to the number of vertices. In this paper, firstly, we give the A α -characteristic polynomial of the coalescent graph, and A α -eigenvalues of the star graph for the application. Secondly, we study the extremal graphs with the maximum and minimum A α -spectral radius of the unicyclic graph. Finally, we present the extremal graph with the maximum A α -separator of the unicyclic graph and calculate the range of A α -separator of the corresponding extremal graph. [ABSTRACT FROM AUTHOR]